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Real analysis with real applications

Author: Kenneth R Davidson; Allan P Donsig
Publisher: Upper Saddle River, NJ : Prentice Hall, ©2002.
Edition/Format:   Print book : EnglishView all editions and formats
Summary:

Using a progressive but flexible format, this text develops the principles of real analysis and shows how they can be used in a variety of applications. Building on students' knowledge of calculus  Read more...

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Additional Physical Format: Online version:
Davidson, Kenneth R.
Real analysis with real applications.
Upper Saddle River, NJ : Prentice Hall, ©2002
(OCoLC)988917171
Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Kenneth R Davidson; Allan P Donsig
ISBN: 0130416479 9780130416476
OCLC Number: 48053890
Description: xvii, 624 pages : illustrations ; 25 cm
Contents: 1. Background. The Language of Mathematics. Sets and Functions. Calculus. Linear Algebra. The Role of Proofs. Appendix: Equivalence Relations.A. ABSTRACT ANALYSIS. 2. The Real Numbers. An Overview of the Real Numbers. Infinite Decimals. Limits. Basic Properties of Limits. Upper and Lower Bounds. Subsequences. Cauchy Sequences. Appendix: Cardinality.3. Series. Convergent Series. Convergence Tests for Series. The Number e. Absolute and Conditional Convergence.4. The Topology of Rn. n-dimensional Space. Convergence and Completeness in Rn. Closed and Open Subsets of Rn. Compact Sets and the Heine-Borel Theorem.5. Functions. Limits and Continuity. Discontinuous Functions. Properties of Continuous Functions. Compactness and Extreme Values. Uniform Continuity. The Intermediate Value Theorem. Monotone Functions.6. Normed Vector Spaces. Differentiable Functions. The Mean Value Theorem. Riemann Integration. The Fundamental Theorem of Calculus. Wallis's Product and Stirling's Formula. Measure Zero and Lebesgue's Theorem.7. Differentiation and Integration. Definition and Examples. Topology in Normed Spaces. Inner Product Spaces. Orthonormal Sets. Orthogonal Expansions in Inner Product Spaces. Finite-Dimensional Normed Spaces. The LP Norms.8. Limits of Functions. Limits of Functions. Uniform Convergence and Continuity. Uniform Convergence and Integration. Series of Functions. Power Series. Compactness and Subsets of C(K).9. Metric Spaces. Definitions and Examples. Compact Metric Spaces. Complete Metric Spaces. Connectedness. Metric Completion. The LP Spaces and Abstract Integration.B. APPLICATIONS. 10. Approximation by Polynomials. Taylor Series. How Not to Approximate a Function. Bernstein's Proof of the Weierstrass Theorem. Accuracy of Approximation. Existence of Best Approximations. Characterizing Best Approximations. Expansions Using Chebychev Polynomials. Splines. Uniform Approximation by Splines. Appendix: The Stone-Weierstrass Theorem.11. Discrete Dynamical Systems. Fixed Points and the Contraction Principle. Newton's Method. Orbits of a Dynamical System. Periodic Points. Chaotic Systems. Topological Conjugacy. Iterated Function Systems and Fractals.12. Differential Equations. Integral Equations and Contractions. Calculus of Vector-Valued Functions. Differential Equations and Fixed Points. Solutions of Differential Equations. Local Solutions. Linear Differential Equations. Perturbation and Stability of Des. Existence without Uniqueness.13. Fourier Series and Physics. The Steady-State Heat Equation. Formal Solution. Orthogonality Relations. Convergence in the Open Disk. The Poisson Formula. Poisson's Theorem. The Maximum Principle. The Vibrating String (Formal Solution). The Vibrating String (Rigourous Solution). Appendix: The Complex Exponential.14. Fourier Series and Approximation. Least Squares Approximations. The Isoperimetric Problem. The Riemann-Lebesgue Lemma. Pointwise Convergence of Fourier Series. Gibbs's Phenomenon. Cesaro Summation of Fourier Series. Best Approximation by Trig Polynomials. Connections with Polynomial Approximation. Jackson's Theorem and Bernstein's Theorem.15. Wavelets. Introduction. The Haar Wavelet. Multiresolution Analysis. Recovering the Wavelet. Daubechies Wavelets. Existence of the Daubechies Wavelets. Approximations Using Wavelets. The Franklin Wavelet. Riesz Multiresolution Analysis.16. Convexity and Optimization. Convex Sets. Relative Interior. Separation Theorems. Extreme Points. Convex Functions in One Dimension. Convex Functions in Higher Dimensions. Subdifferentials and Directional Derivatives. Tangent and Normal Cones. Constrained Minimization. The Minimax Theorem.References. Index.
Responsibility: Kenneth R. Davidson, Allan P. Donsig.

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